American Institute of Mathematical Sciences

Advanced
July  2007, 1(3): 371-424. doi: 10.3934/jmd.2007.1.371

Unbounded orbits for outer billiards I

Richard Evan Schwartz 1,

1. 

Department of Mathematics, Brown University, Providence, RI 02912, United States

Received  January 2007 Revised  February 2007 Published  April 2007

The question of B.H. Neumann, which dates back to the 1950s, asks if there exists an outer billiards system with an unbounded orbit. We prove that outer billiards for the Penrose kite, the convex quadrilateral from the Penrose tiling, has an unbounded orbit. We also analyze some finer properties of the orbit structure, and in particular produce an uncountable family of unbounded orbits. Our methods relate outer billiards on the Penrose kite to polygon exchange maps, arithmetic dynamics, and self-similar tilings.
Keywords: arithmetic graph., Penrose kite, unbounded orbits, polygon exchange, outer billiards, dual billiards.
Mathematics Subject Classification: Primary: 37E99; Secondary: 52C2.
Citation: Richard Evan Schwartz. Unbounded orbits for outer billiards I. Journal of Modern Dynamics, 2007, 1 (3) : 371-424. doi: 10.3934/jmd.2007.1.371
[1]

Richard Evan Schwartz. Outer billiards on the Penrose kite: Compactification and renormalization. Journal of Modern Dynamics, 2011, 5 (3) : 473-581. doi: 10.3934/jmd.2011.5.473
[2]

Richard Evan Schwartz. Research announcement: unbounded orbits for outer billiards. Electronic Research Announcements, 2007, 14: 1-6. doi: 10.3934/era.2007.14.1
[3]

Daniel Genin. Research announcement: Boundedness of orbits for trapezoidal outer billiards. Electronic Research Announcements, 2008, 15: 71-78. doi: 10.3934/era.2008.15.71
[4]

Richard Evan Schwartz. Outer billiards and the pinwheel map. Journal of Modern Dynamics, 2011, 5 (2) : 255-283. doi: 10.3934/jmd.2011.5.255
[5]

Daniel Genin, Serge Tabachnikov. On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards. Journal of Modern Dynamics, 2007, 1 (2) : 155-173. doi: 10.3934/jmd.2007.1.155
[6]

Vladimir Dragović, Milena Radnović. Pseudo-integrable billiards and arithmetic dynamics. Journal of Modern Dynamics, 2014, 8 (1) : 109-132. doi: 10.3934/jmd.2014.8.109
[7]

Misha Bialy. Maximizing orbits for higher-dimensional convex billiards. Journal of Modern Dynamics, 2009, 3 (1) : 51-59. doi: 10.3934/jmd.2009.3.51
[8]

Michel L. Lapidus, Robert G. Niemeyer. Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3719-3740. doi: 10.3934/dcds.2013.33.3719
[9]

Dmitri Scheglov. Growth of periodic orbits and generalized diagonals for typical triangular billiards. Journal of Modern Dynamics, 2013, 7 (1) : 31-44. doi: 10.3934/jmd.2013.7.31
[10]

Giovanni Forni, Carlos Matheus. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. Journal of Modern Dynamics, 2014, 8 (3&4) : 271-436. doi: 10.3934/jmd.2014.8.271
[11]

W. Patrick Hooper, Richard Evan Schwartz. Billiards in nearly isosceles triangles. Journal of Modern Dynamics, 2009, 3 (2) : 159-231. doi: 10.3934/jmd.2009.3.159
[12]

Serge Tabachnikov. Birkhoff billiards are insecure. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 1035-1040. doi: 10.3934/dcds.2009.23.1035
[13]

Simon Castle, Norbert Peyerimhoff, Karl Friedrich Siburg. Billiards in ideal hyperbolic polygons. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 893-908. doi: 10.3934/dcds.2011.29.893
[14]

Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319
[15]

Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048
[16]

Hong-Kun Zhang. Free path of billiards with flat points. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4445-4466. doi: 10.3934/dcds.2012.32.4445
[17]

Giovanni Panti. Billiards on pythagorean triples and their Minkowski functions. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4341-4378. doi: 10.3934/dcds.2020183
[18]

W. Patrick Hooper, Richard Evan Schwartz. Erratum: Billiards in nearly isosceles triangles. Journal of Modern Dynamics, 2014, 8 (1) : 133-137. doi: 10.3934/jmd.2014.8.133
[19]

Pedro Duarte, José Pedro GaivÃo, Mohammad Soufi. Hyperbolic billiards on polytopes with contracting reflection laws. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3079-3109. doi: 10.3934/dcds.2017132
[20]

Misha Bialy. On Totally integrable magnetic billiards on constant curvature surface. Electronic Research Announcements, 2012, 19: 112-119. doi: 10.3934/era.2012.19.112

2019 Impact Factor: 0.465

Tools

Metrics

  • PDF downloads (49)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences